Taiwanese Journal of Mathematics


Shaolei Ru

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We study multilinear operators $T(f_{1},f_{2},...,f_{m})$ that commutes with simultaneous translations and prove that if T is bounded from $L^{p_{1}} \times L^{p_{2}} \times ... \times L^{p_{m}}$ to $L^{p}$, then for any $r \geqslant p$, $0 \lt p,q \leqslant \infty$ and \[s \gt \left\{  \begin{array}{lll}  n(1-1\wedge\frac{1}{q}),     &(\frac{1}{p},\frac{1}{q})\in D_{1};\\  n(1\vee\frac{1}{p}\vee\frac{1}{q}-\frac{1}{q}),    &(\frac{1}{p},\frac{1}{q})\in  \mathbb{R}_{+}^{2}-D_{1},  \end{array}  \right.\] ($D_{1}=\{(\frac{1}{p},\frac{1}{q})\in\mathbb{R}_{+}^{2}:\frac{1}{q}\geqslant\frac{2}{p},\frac{1}{p}\leqslant\frac{1}{2}\}$)T is bounded from $M_{p_{1},q}^{s}\times M_{p_{2},q}^{s}\times...\times M_{p_{m},q}^{s}$ to $M_{r,q}^{s}$ (which improves the results obtained by [5], [6].), where $M_{p,q}^{s}$ is the modulation spaces. Besides, we alsoobtain the similar results for Triebel-type spaces $N_{p,q}^{s}$ introduced by [21] (T is bounded from $N_{p,q}^{s} \times N_{p,q}^{s} \times ... \times N_{p,q}^{s}$ to $N_{p,q}^{s}$). As applications, we obtain the boundedness on the modulation spaces for the bilinear Hilbert transform, bilinear fractional integral, the pointwise product of functions, and the bilinear oscillatory integral along parabolas. Also, in modulation spaces and $N_{p,q}^{s}$, we study the well-posedness of the Cauchy problem for the fractional heat and Schrödinger equations with some new nonlinear terms. Such nonlinear well-posedness problems are not studied in other function spaces.

Article information

Taiwanese J. Math., Volume 18, Number 4 (2014), 1129-1149.

First available in Project Euclid: 10 July 2017

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Zentralblatt MATH identifier

Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B37: Harmonic analysis and PDE [See also 35-XX] 42B35: Function spaces arising in harmonic analysis

multilinear operators modulation spaces Triebel-type spaces well-posedness


Ru, Shaolei. MULTILINEAR ESTIMATES ON FREQUENCY-UNIFORM DECOMPOSITION SPACES AND APPLICATIONS. Taiwanese J. Math. 18 (2014), no. 4, 1129--1149. doi:10.11650/tjm.18.2014.3159. https://projecteuclid.org/euclid.twjm/1499706481

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  • A. Benyi, K. Gröchenig, K. A. Okoudjou and L. G. Rogers, Unimodular Fourier multipliers for modulation spaces, Journal of Functional Analysis, 246 (2007), 366-384.
  • T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^{s}$, Nonlinear Anal. TMA, 14 (1990), 807-836.
  • T. Cazenave, Semilinear Schrödinger Equations, American Mathematical Society, 2004.
  • T. Cazenave and F. B. Weissler, Some Remarks on the Nonlinear Schrödinger Equation in the Critical Case, Nonlinear semigroups, partial differential equations and attractors, (Washington D.C., 1987), Lecture Notes in Mathematics, 1394, Springer, Berlin, 1989, pp. 18-29.
  • J. Chen and X. M. Wu, Boundedness of Fractional Integral Operators on $\alpha$-modulation Spaces, preprint.
  • J. Chen, D. Fan and Y. Fan, Multilinear estimates on modulation spaces and applications, To appear in Applied Math. A Journal of Chinese Universities.
  • J. Chen and D. Fan, Some bilinear eatimates, J. Korean Math. J., 46 (2009), 609-622.
  • J. Chen, Y. Ding, Q. Deng and D. Fan, Estimates on fractional power dissipative equations in function spaces, Nonlinear Analysis: Theory, Methods and Applications, 75 (2012), 2959-2974.
  • D. Fan and S. Sato, Transference on certain multilinear multiplier operators, J. Austral. Math. Soc., 70 (2001), 37-55.
  • Y. Fan and G. L. Gao, Some estimates of rough bilinear fractional integral, J. Function Spaces and Applications, Vol. 2012, ID 406540.
  • H. G. Feichtinger, Modulation spaces on locally Abelian groups, Technical Report, University of Vienna, 1983, Updated version appeared in Proceedings of “International Conference on Wavelets and Applications,” 2002, pp. 99-140, Chennai, India, 2003.
  • J. Gilbert and A. Nahmod, Boundedness of bilinear operators with nonsmooth symbols, Math Research Letters, 7 (2000), 767-778.
  • L. Grafakos, On multilinear fractional Integrals, Studia Math., 102 (1992), 49-56.
  • L. Grafakos, Classical and Modern Fourier Analysis, Pearson/Prentice Hall, 2004.
  • J. S. Han and B. X. Wang, $\alpha$-Modulation Spaces $($I$)($II$)$, arXiv: 1108.0460v3 [math.FA] 16 Feb 2012.
  • M. Hiber and J. Prüss, Heat kernels and maximal $L^{p}$-$L^{q}$ estimates for parabolic evolution equations, Comm. Partial Differential Equations, 22 (1997), 1647-1669.
  • T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincare Phys. Theor., 46 (1987), 113-129.
  • T. Kato, Nonlinear Schrödinger equations, Schrödinger operators (Sonderborg, 1988), pp. 218-163, Lecture Notes in physics, 345, Springer, Berlin, 1989.
  • C. E. Kenig and E. M. Stein, Multilinear estimates and fractional integration, Math. Research Letter, 6 (1999), 1-15.
  • M. Lacey and C. Thiele, $L^{p}$ estimate on the bilinear Hilbert transform, Ann. of Math., 146 (1997), 693-724.
  • S. L. Ru and J. C. Chen, The Well-posedness of Nonlinear Schrödinger Equations in Triebel-type Spaces, submitted.
  • C. Thiele, Multilinear singular integrals, Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations, (EI Escrial, 2000), Publ. Math. 2002, Vol. Extra, 229-274.
  • E. Terraneo, Non-uniquness for a critical nonlinear heat equation, Communications in Partial Differential Equations, 27 (2002), 185-218.
  • H. Triebel, Function Space Theory, Birkhauser-Verlag, 1983.
  • N. Wiener, Tauberian theorems, Ann. of Math., 33 (1932), 1-100.
  • B. X. Wang and H. Hudzik, The global Cauchy problem for the NLS and NLKG with small rough date, J. Differential Equations, 231 (2007), 36-73.
  • B. X. Wang, L. F. Zhao and B. L. Guo, Isometric decompositin operators, function spaces $E_{p,q}^{\lambda}$ and their applications to nonlinear evolution equations, J. Funct. Anal., 233 (2006), 1-39.
  • B. X. Wang and C. Y. Huang, Frequency-uniform decomposition method for the generalized BO, KdV and NLS equations, J. Differential Equations, 239 (2007), 213-250.
  • B. X. Wang, L. J. Han and C. Y. Huang, Global well-posedness and scattering for the derivative nonlinear Schrödinger equation with small rough date, Ann. I. H. Poincare, AN, 26 (2009), to appear.
  • B. X. Wang, C. C. Hao and Z. H. Huo, Introduction of nonlinear evolution equation (in Chinese), to appear.
  • B. X. Wang, The Cauchy problem for the nonlinear Schrödinger equation, nonlinear Klein-Gordon equation and their coupled equations, Doctoral Thesis, Inst. Appl. Phys. and Comput. Math, Dec, 1993, pp. 1-224.
  • Z. Zhai, Strichartz type estimates for fractional heat equations, J. Math. Anal. Appl., 356 (2009), 642-658.